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isaafp
afp2020
Commits
89ed1831f6cd
Commit
0c8f8363
authored
Dec 28, 2020
by
Lawrence Paulson
Browse files
New (ZF!) entry Delta_System_Lemma
parent
a80f4e41ac2b
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thys/Delta_System_Lemma/Cohen_Posets.thy
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subsection
\<
open
>
Application
to
Cohen
posets
\
label
{
sec
:
cohen
}\<
close
>
theory
Cohen_Posets
imports
Delta_System
begin
text
\<
open
>
We
end
this
session
by
applying
DSL
to
the
combinatorics
of
finite
function
posets
.
We
first
define
some
basic
concepts
;
we
take
a
different
approach
from
\
cite
{
2020
arXiv200109715G
},
in
that
the
order
relation
is
presented
as
a
predicate
(
of
type
@{
typ
\<
open
>[
i
,
i
]
\<
Rightarrow
>
o
\<
close
>}).
Two
elements
of
a
poset
are
\<^
emph
>\<
open
>
compatible
\<
close
>
if
they
have
a
common
lower
bound
.\<
close
>
definition
compat_in
::
"[i,[i,i]\<Rightarrow>o,i,i]\<Rightarrow>o"
where
"compat_in(A,r,p,q) \<equiv> \<exists>d\<in>A . r(d,p) \<and> r(d,q)"
text
\<
open
>
An
\<^
emph
>\<
open
>
antichain
\<
close
>
is
a
subset
of
pairwise
incompatible
members
.\<
close
>
definition
antichain
::
"[i,[i,i]\<Rightarrow>o,i]\<Rightarrow>o"
where
"antichain(P,leq,A) \<equiv> A\<subseteq>P \<and> (\<forall>p\<in>A. \<forall>q\<in>A.
p\<noteq>q \<longrightarrow> \<not>compat_in(P,leq,p,q))"
text
\<
open
>
A
poset
has
the
\<^
emph
>\<
open
>
countable
chain
condition
\<
close
>
(
ccc
)
if
all
of
its
antichains
are
countable
.\<
close
>
definition
ccc
::
"[i,[i,i]\<Rightarrow>o]\<Rightarrow>o"
where
"ccc(P,leq) \<equiv> \<forall>A. antichain(P,leq,A) \<longrightarrow> countable(A)"
text
\<
open
>
Finally
,
the
\<^
emph
>\<
open
>
Cohen
poset
\<
close
>
is
the
set
of
finite
partial
functions
between
two
sets
with
the
order
of
reverse
inclusion
.\<
close
>
definition
Fn
::
"[i,i] \<Rightarrow> i"
where
"Fn(I,J) \<equiv> \<Union>{(d\<rightarrow>J) . d \<in> {x \<in> Pow(I). Finite(x)}}"
abbreviation
Supset
::
"i \<Rightarrow> i \<Rightarrow> o"
(
infixl
\<
open
>\<
supseteq
>\<
close
>
50
)
where
"f \<supseteq> g \<equiv> g \<subseteq> f"
lemma
FnI
[
intro
]:
assumes
"p : d \<rightarrow> J"
"d \<subseteq> I"
"Finite(d)"
shows
"p \<in> Fn(I,J)"
using
assms
unfolding
Fn_def
by
auto
lemma
FnD
[
dest
]:
assumes
"p \<in> Fn(I,J)"
shows
"\<exists>d. p : d \<rightarrow> J \<and> d \<subseteq> I \<and> Finite(d)"
using
assms
unfolding
Fn_def
by
auto
lemma
Fn_is_function
:
"p \<in> Fn(I,J) \<Longrightarrow> function(p)"
unfolding
Fn_def
using
fun_is_function
by
auto
lemma
restrict_eq_imp_compat
:
assumes
"f \<in> Fn(I, J)"
"g \<in> Fn(I, J)"
"restrict(f, domain(f) \<inter> domain(g)) = restrict(g, domain(f) \<inter> domain(g))"
shows
"f \<union> g \<in> Fn(I, J)"
proof

from
assms
obtain
d1
d2
where
"f : d1 \<rightarrow> J"
"d1 \<in> Pow(I)"
"Finite(d1)"
"g : d2 \<rightarrow> J"
"d2 \<in> Pow(I)"
"Finite(d2)"
by
blast
with
assms
show
?
thesis
using
domain_of_fun
restrict_eq_imp_Un_into_Pi
[
of
f
d1
"\<lambda>_. J"
g
d2
"\<lambda>_. J"
]
by
auto
qed
text
\<
open
>
We
finally
arrive
to
our
application
of
DSL
.\<
close
>
lemma
ccc_Fn_nat
:
"ccc(Fn(I,2), (\<supseteq>))"
proof

{
fix
A
assume
"\<not> countable(A)"
assume
"A \<subseteq> Fn(I, 2)"
moreover
from
this
have
"countable({p\<in>A. domain(p) = d})"
for
d
proof
(
cases
"Finite(d) \<and> d \<subseteq> I"
)
case
True
with
\<
open
>
A
\<
subseteq
>
Fn
(
I
,
2
)\<
close
>
have
"{p \<in> A . domain(p) = d} \<subseteq> d \<rightarrow> 2"
using
domain_of_fun
by
fastforce
moreover
from
True
have
"Finite(d \<rightarrow> 2)"
using
Finite_Pi
lesspoll_nat_is_Finite
by
auto
ultimately
show
?
thesis
using
subset_Finite
[
of
_
"d\<rightarrow>2"
]
Finite_imp_countable
by
auto
next
case
False
with
\<
open
>
A
\<
subseteq
>
Fn
(
I
,
2
)\<
close
>
have
"{p \<in> A . domain(p) = d} = 0"
by
(
intro
equalityI
)
(
auto
dest
!: domain_of_fun)
then
show
?
thesis
using
empty_lepollI
by
auto
qed
moreover
have
"uncountable({domain(p) . p \<in> A})"
proof
from
\<
open
>
A
\<
subseteq
>
Fn
(
I
,
2
)\<
close
>
have
"A = (\<Union>d\<in>{domain(p) . p \<in> A}. {p\<in>A. domain(p) = d})"
by
auto
moreover
assume
"countable({domain(p) . p \<in> A})"
moreover
note
\<
open
>\<
And
>
d
.
countable
({
p
\<
in
>
A
.
domain
(
p
)
=
d
})\<
close
>
\<
open
>\<
not
>
countable
(
A
)\<
close
>
ultimately
show
"False"
using
countable_imp_countable_UN
[
of
"{domain(p). p\<in>A}"
"\<lambda>d. {p \<in> A. domain(p) = d }"
]
by
auto
qed
moreover
from
\<
open
>
A
\<
subseteq
>
Fn
(
I
,
2
)\<
close
>
have
"p \<in> A \<Longrightarrow> Finite(domain(p))"
for
p
using
lesspoll_nat_is_Finite
[
of
"domain(p)"
]
domain_of_fun
[
of
p
_
"\<lambda>_. 2"
]
by
auto
ultimately
obtain
D
where
"delta_system(D)"
"D \<subseteq> {domain(p) . p \<in> A}"
"D \<approx> \<aleph>\<^bsub>1\<^esub>"
using
delta_system_uncountable
[
of
"{domain(p) . p \<in> A}"
]
by
auto
then
have
delta
:
"\<forall>d1\<in>D. \<forall>d2\<in>D. d1 \<noteq> d2 \<longrightarrow> d1 \<inter> d2 = \<Inter>D"
using
delta_system_root_eq_Inter
by
simp
moreover
from
\<
open
>
D
\<
approx
>
\<
aleph
>\<^
bsub
>
1
\<^
esub
>\<
close
>
have
"uncountable(D)"
using
uncountable_iff_subset_eqpoll_Aleph1
by
auto
moreover
from
this
and
\<
open
>
D
\<
subseteq
>
{
domain
(
p
)
.
p
\<
in
>
A
}\<
close
>
obtain
p1
where
"p1 \<in> A"
"domain(p1) \<in> D"
using
uncountable_not_empty
[
of
D
]
by
blast
moreover
from
this
and
\<
open
>
p1
\<
in
>
A
\<
Longrightarrow
>
Finite
(
domain
(
p1
))\<
close
>
have
"Finite(domain(p1))"
using
Finite_domain
by
simp
moreover
define
r
where
"r \<equiv> \<Inter>D"
ultimately
have
"Finite(r)"
using
subset_Finite
[
of
"r"
"domain(p1)"
]
by
auto
have
"countable({restrict(p,r) . p\<in>A})"
proof

have
"f \<in> Fn(I, 2) \<Longrightarrow> restrict(f,r) \<in> Pow(r \<times> 2)"
for
f
using
restrict_subset_Sigma
[
of
f
_
"\<lambda>_. 2"
r
]
by
(
auto
dest
!:FnD simp: Pi_def) auto
with
\<
open
>
A
\<
subseteq
>
Fn
(
I
,
2
)\<
close
>
have
"{restrict(f,r) . f \<in> A } \<subseteq> Pow(r \<times> 2)"
by
fast
with
\<
open
>
Finite
(
r
)\<
close
>
show
?
thesis
using
Finite_Sigma
[
THEN
Finite_Pow
,
of
r
"\<lambda>_. 2"
]
by
(
intro
Finite_imp_countable
)
(
auto
intro
:
subset_Finite
)
qed
moreover
have
"uncountable({p\<in>A. domain(p) \<in> D})"
(
is
"uncountable(?X)"
)
proof
from
\<
open
>
D
\<
subseteq
>
{
domain
(
p
)
.
p
\<
in
>
A
}\<
close
>
have
"(\<lambda>p\<in>?X. domain(p)) \<in> surj(?X, D)"
using
lam_type
unfolding
surj_def
by
auto
moreover
assume
"countable(?X)"
moreover
note
\<
open
>
uncountable
(
D
)\<
close
>
ultimately
show
False
using
surj_countable
by
auto
qed
moreover
have
"D = (\<Union>f\<in>Pow(r\<times>2) . {domain(p) . p\<in>{ x\<in>A. restrict(x,r) = f \<and> domain(x) \<in> D}})"
proof

{
fix
z
assume
"z \<in> D"
with
\<
open
>
D
\<
subseteq
>
_
\<
close
>
obtain
p
where
"domain(p) = z"
"p \<in> A"
by
auto
moreover
from
\<
open
>
A
\<
subseteq
>
Fn
(
I
,
2
)\<
close
>
and
this
have
"p : z \<rightarrow> 2"
using
domain_of_fun
by
(
auto
dest
!:FnD)
moreover
from
this
have
"restrict(p,r) \<subseteq> r \<times> 2"
using
function_restrictI
[
of
p
r
]
fun_is_function
[
of
p
z
"\<lambda>_. 2"
]
restrict_subset_Sigma
[
of
p
z
"\<lambda>_. 2"
r
]
by
(
auto
simp
:
Pi_def
)
ultimately
have
"\<exists>p\<in>A. restrict(p,r) \<in> Pow(r\<times>2) \<and> domain(p) = z"
by
auto
}
then
show
?
thesis
by
(
intro
equalityI
)
(
force
)+
qed
obtain
f
where
"uncountable({domain(p) . p\<in>{x\<in>A. restrict(x,r) = f \<and> domain(x) \<in> D}})"
(
is
"uncountable(?Y(f))"
)
proof

{
from
\<
open
>
Finite
(
r
)\<
close
>
have
"countable(Pow(r\<times>2))"
using
Finite_Sigma
[
THEN
Finite_Pow
,
THEN
Finite_imp_countable
]
by
simp
moreover
assume
"countable(?Y(f))"
for
f
moreover
note
\<
open
>
D
=
(\<
Union
>
f
\<
in
>
Pow
(
r
\<
times
>
2
)
.?
Y
(
f
))\<
close
>
moreover
note
\<
open
>
uncountable
(
D
)\<
close
>
ultimately
have
"False"
using
countable_imp_countable_UN
[
of
"Pow(r\<times>2)"
?
Y
]
by
auto
}
with
that
show
?
thesis
by
auto
qed
then
obtain
j
where
"j \<in> inj(nat, ?Y(f))"
using
uncountable_iff_nat_lt_cardinal
[
THEN
iffD1
,
THEN
leI
,
THEN
cardinal_le_imp_lepoll
,
THEN
lepollD
]
by
auto
then
have
"j`0 \<noteq> j`1"
"j`0 \<in> ?Y(f)"
"j`1 \<in> ?Y(f)"
using
inj_is_fun
[
THEN
apply_type
,
of
j
nat
"?Y(f)"
]
unfolding
inj_def
by
auto
then
obtain
p
q
where
"domain(p) \<noteq> domain(q)"
"p \<in> A"
"q \<in> A"
"domain(p) \<in> D"
"domain(q) \<in> D"
"restrict(p,r) = restrict(q,r)"
by
auto
moreover
from
this
and
delta
have
"domain(p) \<inter> domain(q) = r"
unfolding
r_def
by
simp
moreover
note
\<
open
>
A
\<
subseteq
>
Fn
(
I
,
2
)\<
close
>
moreover
from
calculation
have
"p \<union> q \<in> Fn(I, 2)"
by
(
rule_tac
restrict_eq_imp_compat
)
auto
ultimately
have
"\<exists>p\<in>A. \<exists>q\<in>A. p \<noteq> q \<and> compat_in(Fn(I, 2), (\<supseteq>), p, q)"
unfolding
compat_in_def
by
(
rule_tac
bexI
[
of
_
p
],
rule_tac
bexI
[
of
_
q
])
blast
}
then
show
?
thesis
unfolding
ccc_def
antichain_def
by
auto
qed
text
\<
open
>
The
fact
that
a
poset
$
P
$
has
the
ccc
has
useful
consequences
for
the
theory
of
forcing
,
since
it
implies
that
cardinals
from
the
original
model
are
exactly
the
cardinals
in
any
generic
extension
by
$
P
$
\
cite
[
Chap
.~
IV
]{
kunen2011set
}.\<
close
>
end
\ No newline at end of file
thys/Delta_System_Lemma/Delta_System.thy
0 → 100644
View file @
89ed1831
section\<open>The Delta System Lemma\label{sec:dsl}\<close>
theory Delta_System
imports
Cardinal_Library
begin
text\<open>A \<^emph>\<open>delta system\<close> is family of sets with a common pairwise
intersection.\<close>
definition
delta_system :: "i \<Rightarrow> o" where
"delta_system(D) \<equiv> \<exists>r. \<forall>A\<in>D. \<forall>B\<in>D. A \<noteq> B \<longrightarrow> A \<inter> B = r"
lemma delta_systemI[intro]:
assumes "\<forall>A\<in>D. \<forall>B\<in>D. A \<noteq> B \<longrightarrow> A \<inter> B = r"
shows "delta_system(D)"
using assms unfolding delta_system_def by simp
lemma delta_systemD[dest]:
"delta_system(D) \<Longrightarrow> \<exists>r. \<forall>A\<in>D. \<forall>B\<in>D. A \<noteq> B \<longrightarrow> A \<inter> B = r"
unfolding delta_system_def by simp
text\<open>Hence, pairwise intersections equal the intersection of the whole
family.\<close>
lemma delta_system_root_eq_Inter:
assumes "delta_system(D)"
shows "\<forall>A\<in>D. \<forall>B\<in>D. A \<noteq> B \<longrightarrow> A \<inter> B = \<Inter>D"
proof (clarify, intro equalityI, auto)
fix A' B' x C
assume hyp:"A'\<in>D" "B'\<in> D" "A'\<noteq>B'" "x\<in>A'" "x\<in>B'" "C\<in>D"
with assms
obtain r where delta:"\<forall>A\<in>D. \<forall>B\<in>D. A \<noteq> B \<longrightarrow> A \<inter> B = r"
by auto
show "x \<in> C"
proof (cases "C=A'")
case True
with hyp and assms
show ?thesis by simp
next
case False
moreover
note hyp
moreover from calculation and delta
have "r = C \<inter> A'" "A' \<inter> B' = r" "x\<in>r" by auto
ultimately
show ?thesis by simp
qed
qed
text\<open>The \<^emph>\<open>Delta System Lemma\<close> (DSL) states that any uncountable family of
finite sets includes an uncountable delta system. This is the simplest
non trivial version; others, for cardinals greater than \<^term>\<open>\<aleph>\<^bsub>1\<^esub>\<close> assume
some weak versions of the generalized continuum hypothesis for the
cardinals involved.
The proof is essentially the one in \cite[III.2.6]{kunen2011set} for the
case \<^term>\<open>\<aleph>\<^bsub>1\<^esub>\<close>; another similar presentation can be found in
\cite[Chap.~16]{JW}.\<close>
lemma delta_system_Aleph1:
assumes "\<forall>A\<in>F. Finite(A)" "F \<approx> \<aleph>\<^bsub>1\<^esub>"
shows "\<exists>D. D \<subseteq> F \<and> delta_system(D) \<and> D \<approx> \<aleph>\<^bsub>1\<^esub>"
proof 
text\<open>Since all members are finite,\<close>
from \<open>\<forall>A\<in>F. Finite(A)\<close>
have "(\<lambda>A\<in>F. A) : F \<rightarrow> \<omega>" (is "?cards : _")
by (rule_tac lam_type) simp
moreover from this
have a:"?cards `` {n} = { A\<in>F . A = n }" for n
using vimage_lam by auto
moreover
note \<open>F \<approx> \<aleph>\<^bsub>1\<^esub>\<close>
moreover from calculation
text\<open>there are uncountably many have the same cardinal:\<close>
obtain n where "n\<in>\<omega>" "?cards `` {n} = \<aleph>\<^bsub>1\<^esub>"
using eqpoll_Aleph1_cardinal_vimage[of F ?cards] by auto
moreover
define G where "G \<equiv> ?cards `` {n}"
moreover from calculation
have "G \<subseteq> F" by auto
ultimately
text\<open>Therefore, without loss of generality, we can assume that all
elements of the family have cardinality \<^term>\<open>n\<in>\<omega>\<close>.\<close>
have "A\<in>G \<Longrightarrow> A = n" and "G \<approx> \<aleph>\<^bsub>1\<^esub>" for A
using cardinal_Card_eqpoll_iff by auto
with \<open>n\<in>\<omega>\<close>
text\<open>So we prove the result by induction on this \<^term>\<open>n\<close> and
generalizing \<^term>\<open>G\<close>, since the argument requires changing the
family in order to apply the inductive hypothesis.\<close>
have "\<exists>D. D \<subseteq> G \<and> delta_system(D) \<and> D \<approx> \<aleph>\<^bsub>1\<^esub>"
proof (induct arbitrary:G)
case 0 \<comment> \<open>This case is impossible\<close>
then
have "G \<subseteq> {0}"
using cardinal_0_iff_0 by auto
with \<open>G \<approx> \<aleph>\<^bsub>1\<^esub>\<close>
show ?case
using nat_lt_Aleph1 subset_imp_le_cardinal[of G "{0}"]
lt_trans2 cardinal_Card_eqpoll_iff by auto
next
case (succ n)
then
have "\<forall>a\<in>G. Finite(a)"
using Finite_cardinal_iff' nat_into_Finite[of "succ(n)"]
by fastforce
show "\<exists>D. D \<subseteq> G \<and> delta_system(D) \<and> D \<approx> \<aleph>\<^bsub>1\<^esub>"
proof (cases "\<exists>p. {A\<in>G . p \<in> A} \<approx> \<aleph>\<^bsub>1\<^esub>")
case True \<comment> \<open>the positive case, uncountably many sets with a
common element\<close>
then
obtain p where "{A\<in>G . p \<in> A} \<approx> \<aleph>\<^bsub>1\<^esub>" by blast
moreover from this
have "{A{p} . A\<in>{X\<in>G. p\<in>X}} \<approx> \<aleph>\<^bsub>1\<^esub>" (is "?F \<approx> _")
using Diff_bij[of "{A\<in>G . p \<in> A}" "{p}"]
comp_bij[OF bij_converse_bij, where C="\<aleph>\<^bsub>1\<^esub>"] by fast
text\<open>Now using the hypothesis of the successor case,\<close>
moreover from \<open>\<And>A. A\<in>G \<Longrightarrow> A=succ(n)\<close> \<open>\<forall>a\<in>G. Finite(a)\<close>
and this
have "p\<in>A \<Longrightarrow> A\<in>G \<Longrightarrow> A  {p} = n" for A
using Finite_imp_succ_cardinal_Diff[of _ p] by force
moreover from this and \<open>n\<in>\<omega>\<close>
have "\<forall>a\<in>?F. Finite(a)"
using Finite_cardinal_iff' nat_into_Finite by auto
moreover
text\<open>we may apply the inductive hypothesis to the new family \<^term>\<open>?F\<close>:\<close>
note \<open>(\<And>A. A \<in> ?F \<Longrightarrow> A = n) \<Longrightarrow> ?F \<approx> \<aleph>\<^bsub>1\<^esub> \<Longrightarrow>
\<exists>D. D \<subseteq> ?F \<and> delta_system(D) \<and> D \<approx> \<aleph>\<^bsub>1\<^esub>\<close>
ultimately
obtain D where "D\<subseteq>{A{p} . A\<in>{X\<in>G. p\<in>X}}" "delta_system(D)" "D \<approx> \<aleph>\<^bsub>1\<^esub>"
by auto
moreover from this
obtain r where "\<forall>A\<in>D. \<forall>B\<in>D. A \<noteq> B \<longrightarrow> A \<inter> B = r"
by fastforce
then
have "\<forall>A\<in>D.\<forall>B\<in>D. A\<union>{p} \<noteq> B\<union>{p}\<longrightarrow>(A \<union> {p}) \<inter> (B \<union> {p}) = r\<union>{p}"
by blast
ultimately
have "delta_system({B \<union> {p} . B\<in>D})" (is "delta_system(?D)")
by fastforce
moreover from \<open>D \<approx> \<aleph>\<^bsub>1\<^esub>\<close>
have "D = \<aleph>\<^bsub>1\<^esub>" "Infinite(D)"
using cardinal_eqpoll_iff
by (auto intro!: uncountable_iff_subset_eqpoll_Aleph1[THEN iffD2]
uncountable_imp_Infinite) force
moreover from this
have "?D \<approx> \<aleph>\<^bsub>1\<^esub>"
using cardinal_map_Un[of D "{p}"] naturals_lt_nat
cardinal_eqpoll_iff[THEN iffD1] by simp
moreover
note \<open>D \<subseteq> {A{p} . A\<in>{X\<in>G. p\<in>X}}\<close>
have "?D \<subseteq> G"
proof 
{
fix A
assume "A\<in>G" "p\<in>A"
moreover from this
have "A = A  {p} \<union> {p}"
by blast
ultimately
have "A {p} \<union> {p} \<in> G"
by auto
}
with \<open>D \<subseteq> {A{p} . A\<in>{X\<in>G. p\<in>X}}\<close>
show ?thesis
by blast
qed
ultimately
show "\<exists>D. D \<subseteq> G \<and> delta_system(D) \<and> D \<approx> \<aleph>\<^bsub>1\<^esub>" by auto
next
case False
note \<open>\<not> (\<exists>p. {A \<in> G . p \<in> A} \<approx> \<aleph>\<^bsub>1\<^esub>)\<close> \<comment> \<open>the other case\<close>
moreover from \<open>G \<approx> \<aleph>\<^bsub>1\<^esub>\<close>
have "{A \<in> G . p \<in> A} \<lesssim> \<aleph>\<^bsub>1\<^esub>" (is "?G(p) \<lesssim> _") for p
by (blast intro:lepoll_eq_trans[OF subset_imp_lepoll])
ultimately
have "?G(p) \<prec> \<aleph>\<^bsub>1\<^esub>" for p
unfolding lesspoll_def by simp
then (* may omit the previous step if unfolding here: *)
have "?G(p) \<lesssim> \<omega>" for p
using lesspoll_aleph_plus_one[of 0] Aleph_zero_eq_nat by auto
moreover
have "{A \<in> G . S \<inter> A \<noteq> 0} = (\<Union>p\<in>S. ?G(p))" for S
by auto
ultimately
have "countable(S) \<Longrightarrow> countable({A \<in> G . S \<inter> A \<noteq> 0})" for S
using InfCard_nat Card_nat
le_Card_iff[THEN iffD2, THEN [3] leqpoll_imp_cardinal_UN_le,
THEN [2] le_Card_iff[THEN iffD1], of \<omega> S]
unfolding countable_def by simp
text\<open>For every countable subfamily of \<^term>\<open>G\<close> there is another some
element disjoint from all of them:\<close>
have "\<exists>A\<in>G. \<forall>S\<in>X. S \<inter> A = 0" if "X < \<aleph>\<^bsub>1\<^esub>" "X \<subseteq> G" for X
proof 
from \<open>n\<in>\<omega>\<close> \<open>\<And>A. A\<in>G \<Longrightarrow> A = succ(n)\<close>
have "A\<in>G \<Longrightarrow> Finite(A)" for A
using cardinal_Card_eqpoll_iff
unfolding Finite_def by fastforce
with \<open>X\<subseteq>G\<close>
have "A\<in>X \<Longrightarrow> countable(A)" for A
using Finite_imp_countable by auto
with \<open>X < \<aleph>\<^bsub>1\<^esub>\<close>
have "countable(\<Union>X)"
using Card_nat[THEN cardinal_lt_csucc_iff, of X]
countable_union_countable countable_iff_cardinal_le_nat
unfolding Aleph_def by simp
with \<open>countable(_) \<Longrightarrow> countable({A \<in> G . _ \<inter> A \<noteq> 0})\<close>
have "countable({A \<in> G . (\<Union>X) \<inter> A \<noteq> 0})" .
with \<open>G \<approx> \<aleph>\<^bsub>1\<^esub>\<close>
obtain B where "B\<in>G" "B \<notin> {A \<in> G . (\<Union>X) \<inter> A \<noteq> 0}"
using nat_lt_Aleph1 cardinal_Card_eqpoll_iff[of "\<aleph>\<^bsub>1\<^esub>" G]
uncountable_not_subset_countable[of "{A \<in> G . (\<Union>X) \<inter> A \<noteq> 0}" G]
uncountable_iff_nat_lt_cardinal
by auto
then
show "\<exists>A\<in>G. \<forall>S\<in>X. S \<inter> A = 0" by auto
qed
moreover from \<open>G \<approx> \<aleph>\<^bsub>1\<^esub>\<close>
obtain b where "b\<in>G"
using uncountable_iff_subset_eqpoll_Aleph1
uncountable_not_empty by blast
ultimately
text\<open>Hence, the hypotheses to perform a boundedcardinal selection
are satisfied,\<close>
obtain S where "S:\<aleph>\<^bsub>1\<^esub>\<rightarrow>G" "\<alpha>\<in>\<aleph>\<^bsub>1\<^esub> \<Longrightarrow> \<beta>\<in>\<aleph>\<^bsub>1\<^esub> \<Longrightarrow> \<alpha><\<beta> \<Longrightarrow> S`\<alpha> \<inter> S`\<beta> = 0"
for \<alpha> \<beta>
using bounded_cardinal_selection[of "\<aleph>\<^bsub>1\<^esub>" G "\<lambda>s a. s \<inter> a = 0" b]
by force
then
have "\<alpha> \<in> \<aleph>\<^bsub>1\<^esub> \<Longrightarrow> \<beta> \<in> \<aleph>\<^bsub>1\<^esub> \<Longrightarrow> \<alpha>\<noteq>\<beta> \<Longrightarrow> S`\<alpha> \<inter> S`\<beta> = 0" for \<alpha> \<beta>
using lt_neq_symmetry[of "\<aleph>\<^bsub>1\<^esub>" "\<lambda>\<alpha> \<beta>. S`\<alpha> \<inter> S`\<beta> = 0"] Card_is_Ord
by auto blast
text\<open>and a symmetry argument shows that obtained \<^term>\<open>S\<close> is
an injective \<^term>\<open>\<aleph>\<^bsub>1\<^esub>\<close>sequence of disjoint elements of \<^term>\<open>G\<close>.\<close>
moreover from this and \<open>\<And>A. A\<in>G \<Longrightarrow> A = succ(n)\<close> \<open>S : \<aleph>\<^bsub>1\<^esub> \<rightarrow> G\<close>
have "S \<in> inj(\<aleph>\<^bsub>1\<^esub>, G)"
using cardinal_succ_not_0 Int_eq_zero_imp_not_eq[of "\<aleph>\<^bsub>1\<^esub>" "\<lambda>x. S`x"]
unfolding inj_def by fastforce
moreover from calculation
have "range(S) \<approx> \<aleph>\<^bsub>1\<^esub>"
using inj_bij_range eqpoll_sym unfolding eqpoll_def by fast
moreover from calculation
have "range(S) \<subseteq> G"
using inj_is_fun range_fun_subset_codomain by fast
ultimately
show "\<exists>D. D \<subseteq> G \<and> delta_system(D) \<and> D \<approx> \<aleph>\<^bsub>1\<^esub>"
using inj_is_fun range_eq_image[of S "\<aleph>\<^bsub>1\<^esub>" G]
image_function[OF fun_is_function, OF inj_is_fun, of S "\<aleph>\<^bsub>1\<^esub>" G]
domain_of_fun[OF inj_is_fun, of S "\<aleph>\<^bsub>1\<^esub>" G]
by (rule_tac x="S``\<aleph>\<^bsub>1\<^esub>" in exI) auto
text\<open>This finishes the successor case and hence the proof.\<close>
qed
qed
with \<open>G \<subseteq> F\<close>
show ?thesis by blast
qed
lemma delta_system_uncountable:
assumes "\<forall>A\<in>F. Finite(A)" "uncountable(F)"
shows "\<exists>D. D \<subseteq> F \<and> delta_system(D) \<and> D \<approx> \<aleph>\<^bsub>1\<^esub>"
proof 
from assms
obtain S where "S \<subseteq> F" "S \<approx> \<aleph>\<^bsub>1\<^esub>"
using uncountable_iff_subset_eqpoll_Aleph1[of F] by auto
moreover from \<open>\<forall>A\<in>F. Finite(A)\<close> and this
have "\<forall>A\<in>S. Finite(A)" by auto
ultimately
show ?thesis using delta_system_Aleph1[of S]
by auto
qed
end
\ No newline at end of file
thys/Delta_System_Lemma/Konig.thy
0 → 100644
View file @
89ed1831
theory Konig
imports
Cofinality
Cardinal_Library
begin
text\<open>Now, using the Axiom of choice, we can show that all successor
cardinals are regular.\<close>
lemma cf_csucc:
assumes "InfCard(z)"
shows "cf(z\<^sup>+) = z\<^sup>+"
proof (rule ccontr)
assume "cf(z\<^sup>+) \<noteq> z\<^sup>+"
moreover from \<open>InfCard(z)\<close>
have "Ord(z\<^sup>+)" "Ord(z)" "Limit(z)" "Limit(z\<^sup>+)" "Card(z\<^sup>+)" "Card(z)"
using InfCard_csucc Card_is_Ord InfCard_is_Card InfCard_is_Limit
by fastforce+
moreover from calculation
have "cf(z\<^sup>+) < z\<^sup>+"
using cf_le_cardinal[of "z\<^sup>+", THEN le_iff[THEN iffD1]]
Card_cardinal_eq
by simp
ultimately
obtain G where "G:cf(z\<^sup>+)\<rightarrow> z\<^sup>+" "\<forall>\<beta>\<in>z\<^sup>+. \<exists>y\<in>cf(z\<^sup>+). \<beta> < G`y"
using Limit_cofinal_fun_lt[of "z\<^sup>+" _ "cf(z\<^sup>+)"] Ord_cf
cf_le_cf_fun[of "z\<^sup>+" "cf(z\<^sup>+)"] le_refl[of "cf(z\<^sup>+)"]
by auto
with \<open>Card(z)\<close> \<open>Card(z\<^sup>+)\<close> \<open>Ord(z\<^sup>+)\<close>
have "\<forall>\<beta>\<in>cf(z\<^sup>+). G`\<beta> \<le> z"
using apply_type[of G "cf(z\<^sup>+)" "\<lambda>_. z\<^sup>+", THEN ltI] Card_lt_iff[THEN iffD2]
Ord_in_Ord[OF Card_is_Ord, of "z\<^sup>+"] cardinal_lt_csucc_iff[THEN iffD1]
by auto
from \<open>cf(z\<^sup>+) < z\<^sup>+\<close> \<open>InfCard(z)\<close> \<open>Ord(z)\<close>
have "cf(z\<^sup>+) \<lesssim> z"
using cardinal_lt_csucc_iff[of "z" "cf(z\<^sup>+)"] Card_csucc[of "z"]
le_Card_iff[of "z" "cf(z\<^sup>+)"] InfCard_is_Card
Card_lt_iff[of "cf(z\<^sup>+)" "z\<^sup>+"] lt_Ord[of "cf(z\<^sup>+)" "z\<^sup>+"]
by simp
with \<open>cf(z\<^sup>+) < z\<^sup>+\<close> \<open>\<forall>\<beta>\<in>cf(z\<^sup>+). G`\<beta> \<le> _\<close> \<open>InfCard(z)\<close>
have "\<Union>\<beta>\<in>cf(z\<^sup>+). G`\<beta> \<le> z"
using InfCard_csucc[of z]
subset_imp_lepoll[THEN lepoll_imp_Card_le, of "\<Union>\<beta>\<in>cf(z\<^sup>+). G`\<beta>" "z"]
by (rule_tac leqpoll_imp_cardinal_UN_le) auto
moreover
note \<open>Ord(z)\<close>
moreover from \<open>\<forall>\<beta>\<in>z\<^sup>+. \<exists>y\<in>cf(z\<^sup>+). \<beta> < G`y\<close> and this
have "z\<^sup>+ \<subseteq> (\<Union>\<beta>\<in>cf(z\<^sup>+). G`\<beta>)"
by (blast dest:ltD)
ultimately
have "z\<^sup>+ \<le> z"
using subset_imp_le_cardinal[of "z\<^sup>+" "\<Union>\<beta>\<in>cf(z\<^sup>+). G`\<beta>"] le_trans
InfCard_is_Card Card_csucc[of z] Card_cardinal_eq
by auto
with \<open>Ord(z)\<close>
show "False"
using lt_csucc[of z] not_lt_iff_le[THEN iffD2, of z "z\<^sup>+"]
Card_csucc[THEN Card_is_Ord]
by auto
qed
text\<open>And this finishes the calculation of cofinality of Alephs.\<close>
lemma cf_Aleph_succ: "Ord(z) \<Longrightarrow> cf(\<aleph>\<^bsub>succ(z)\<^esub>) = \<aleph>\<^bsub>succ(z)\<^esub>"
using Aleph_succ cf_csucc InfCard_Aleph by simp
subsection\<open>König's Theorem\label{sec:konig}\<close>
text\<open>We end this section by proving König's Theorem on the cofinality
of cardinal exponentiation. This is a strengthening of Cantor's theorem
and it is essentially the only basic way to prove strict cardinal
inequalities.
It is proved rather straightforwardly with the tools already developed.\<close>
lemma konigs_theorem:
notes [dest] = InfCard_is_Card Card_is_Ord
and [trans] = lt_trans1 lt_trans2
assumes
"InfCard(\<kappa>)" "InfCard(\<nu>)" "cf(\<kappa>) \<le> \<nu>"
shows
"\<kappa> < \<kappa>\<^bsup>\<up>\<nu>\<^esup>"